There isn't anything special in the API that would let you do this. You could send the user directly to the ask a question page. 

I fiddled with the [FireFox plugin][1] I created to see if I could add the ability to post directly to the site to add an answer to a question or a comment if someone was already logged in. If I remember correctly there is some key that is required for the POST request and it wasn't obvious how that key was generated.


  [1]: http://stackapps.com/questions/316/swatch-a-firefox-plugin-for-monitoring-stack-exchange-sites-for-interesting-chan